In response to firstname.lastname@example.org (K. Ann Renninger): You posit that "part of becoming a good math student for some people is learning to speak the language." >>> WHAT language? WHOSE language? This presupposes that there exists an >>>externally-validated mathematical language independent of the users of that >>>language. This is as fallacious as thinking that English exists >>>independently of its users. Clearly there are many variations of English, by >>>country, region, even by family or peer group. Moreover, English is >>>constantly in flux. Likewise, mathematical meanings can greatly vary between >>>groups (just compare definitions given in several texts supposedly >>>addressing the same subject!) and across time (look at early definitions of >>>limit or derivative).
You suggest that "for some students, it may cause confusion to have to speak of Math in everyday English terms." >>>If they can't speak of math in their own terms, then what are they really >>>doing? Do they have any understanding of what they are doing? Is the >>>knowledge really their own or are they just repeating what they have been >>>told? Do they know what they are doing? Take a look at the NCTM Standards, >>>where communication is highlighted as one of the primary goals of >>>mathematics education grades K-12 (and I would argue, beyond)
You ask, "Is there a happy medium where both types of students may be helped?" >>>Yes! The classroom in which students are asked to discuss and talk about >>>their mathematical meanings. The students who are "learning the language" >>>really aren't learning much mathematics. They need help! They need to be >>>transformed into the kind of students who know what they are doing. They may >>>be lost for a while, but that's the breaks. It's good for them in the long >>>run.
By the way, I am not speaking theoretically. I live this every day with a class of high school geometry students (grade 9, heterogeneously grouped) for which we are creating instructional materials (tasks, not a textbook per se). We have been having interesting discussions about how you decide on a definition. If it says so in the dictionary is that good enough? (Probably not, because the dictionary may not have the geometrical meaning of, for example, translation.) What about a math dictionary? (Maybe, but it often does not make much sense.) So students are getting the idea that terms should be understandable; WE should decide what they mean. And, as our knowledge changes, so should our definitions. For example, an early definition of orientation was "which way a figure faces." Which seemed fine to the class, until they got into an argument about a figure rotated 180 degrees--are they facing the same way or not? This discussion lead to a lot of thinking about what orientation is and ended up in a definition that they really understood (well, at least many of them understood). To me, they created some nice mathematics. Memorizing the definition of orientation would not have the same effect.
Finally, I agree that mathematics should not be taught from a well-structured point of view. But not sure completely ill-structured is quite right either. Most important ideas need to be revisited several times over a long period of time to be really understood; cf. van Hiele research. There is just no way that students will catch the richness of quadrilaterals in a two-week unit. They need to just mess around with them, to explore their properties, to look at how different kinds are related, to think about what information will guarantee that a given quad is of a given type, etc. -- the levels of thinking are *very* different in these tasks; they can't be effectively juxtaposed within that proverbial two-week unit. So there is an implied ordering of topics by the level of thinking. However, the content cannot be completely ill-structured either. For example, consider an exploration of sufficient conditions for parallelograms. It would be nice if they had some understanding of triangle congruence postulates, angles of parallel lines, etc.--useful tools for exploring when something is a parallelogram! So we attack ordering of concepts more along the lines of what tools/ways of thinking would we like them to have in order to to develop particular tools for thinking about a given topic at a given point in time. So there may be ordering(s) but not from a rigidly logical point of view.